| L(s) = 1 | + 1.30·2-s − 2.88·3-s − 0.302·4-s + 2.88·5-s − 3.75·6-s − 3·8-s + 5.30·9-s + 3.75·10-s + 5.90·11-s + 0.872·12-s − 8.30·15-s − 3.30·16-s + 0.872·17-s + 6.90·18-s + 2.88·19-s − 0.872·20-s + 7.69·22-s + 6.60·23-s + 8.64·24-s + 3.30·25-s − 6.63·27-s + 1.30·29-s − 10.8·30-s + 0.872·31-s + 1.69·32-s − 17.0·33-s + 1.13·34-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 1.66·3-s − 0.151·4-s + 1.28·5-s − 1.53·6-s − 1.06·8-s + 1.76·9-s + 1.18·10-s + 1.78·11-s + 0.251·12-s − 2.14·15-s − 0.825·16-s + 0.211·17-s + 1.62·18-s + 0.661·19-s − 0.195·20-s + 1.64·22-s + 1.37·23-s + 1.76·24-s + 0.660·25-s − 1.27·27-s + 0.241·29-s − 1.97·30-s + 0.156·31-s + 0.300·32-s − 2.96·33-s + 0.194·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506302906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506302906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 17 | \( 1 - 0.872T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 - 0.872T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 - 7.50T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 9.60T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 5.76T + 73T^{2} \) |
| 79 | \( 1 - 0.605T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 + 7.77T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.35084199689491489552708434131, −6.58414161780023553282556777298, −6.27830942293083491367916344249, −5.64280468255688053675616140784, −5.18232574764156910661665875176, −4.51739259526234953127038261307, −3.81367520020455237225219239822, −2.81026517375331879482562573813, −1.50099855986213935497839866693, −0.823601577912411213656678380839,
0.823601577912411213656678380839, 1.50099855986213935497839866693, 2.81026517375331879482562573813, 3.81367520020455237225219239822, 4.51739259526234953127038261307, 5.18232574764156910661665875176, 5.64280468255688053675616140784, 6.27830942293083491367916344249, 6.58414161780023553282556777298, 7.35084199689491489552708434131
